Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

The axisymmetric linear static mechanics problem under consideration can be solved analytically. The response to the load (volume force and surface force) is solved independently and then summed up.

**Quadratic density force** :math:`{F}_{V}(r)=\alpha {r}^{2}`

We consider the equilibrium equations in cylindrical coordinates:


.. image:: images/Object_3.svg
    :width: 270
    :height: 122


.. _RefImage_Object_3.svg:

which are simplified, given the axial symmetry, by:

.. image:: images/Object_4.svg
    :width: 270
    :height: 122


.. _RefImage_Object_4.svg:

By using the law of behavior and then the deformation-displacement relationships, we arrive at the following differential equation:

.. image:: images/Object_5.svg
    :width: 270
    :height: 122


.. _RefImage_Object_5.svg:

The density force applied is of the type: fV=.r²


.. _Ref497284509:

The solution to the differential equation is then written as:


.. image:: images/Object_6.svg
    :width: 270
    :height: 122


.. _RefImage_Object_6.svg:


.. _RefEquation 2.1-1:

                     eq 2.1-1

The two integration constants :math:`{c}_{1}` and :math:`{c}_{2}` are determined using the boundary conditions:

.. image:: images/Object_7.svg
    :width: 270
    :height: 122


.. _RefImage_Object_7.svg:

We get:

.. image:: images/Object_8.svg
    :width: 270
    :height: 122


.. _RefImage_Object_8.svg:

**Surface force type pressure** :math:`{F}_{S}({R}_{\text{int}})=P`

The problem to be solved is of the same nature, but with zero applied volume force: :math:`{f}_{V}=0` or :math:`\alpha =0`.

The solution on the go [:ref:`éq 2.1-1 <éq 2.1-1>`] is then written:

.. image:: images/Object_9.svg
    :width: 270
    :height: 122


.. _RefImage_Object_9.svg:

, having to respect the conditions:

.. image:: images/Object_10.svg
    :width: 270
    :height: 122


.. _RefImage_Object_10.svg:

This results in:


.. image:: images/Object_11.svg
    :width: 270
    :height: 122


.. _RefImage_Object_11.svg:


.. _RefEquation 2.1-2:

                 Eq 2.1-2


Benchmark results
----------------------

**Digital application:**


.. csv-table::

    "* height", "= :math:`0.5m`;"
    "* inner radius", "= :math:`1m`;"
    "* outer radius", "= :math:`1.4m`;"
    "* :math:`E` ", "= :math:`10\mathrm{Pa}`;"
    "* :math:`\rho` ", "= :math:`1\mathrm{kg}/{m}^{3}`;"
    "* :math:`\nu` ", "= :math:`0.3`;"
    "* :math:`\alpha` ", "= :math:`1N/{m}^{5}`;"
    "* :math:`P` ", "= :math:`1N/{m}^{2}`."


by injecting the numerical values into the solutions [:ref:`éq 2.1-1 <éq 2.1-1>`] and [:ref:`éq 2.1-2 <éq 2.1-2>`] we find after summation:


.. image:: images/Object_12.svg
    :width: 270
    :height: 122


.. _RefImage_Object_12.svg:


Uncertainties about the solution
----------------------------

None (analytical reference solution).